Residual categories of quadric surface bundles

TL;DR

This paper establishes conditions under which residual categories of quadric surface bundles are equivalent to twisted derived categories of schemes, providing new geometric descriptions and applications to complete intersections.

Contribution

It proves the equivalence of residual categories with twisted derived categories under specific hypotheses and offers two geometric proofs for the case with a smooth section.

Findings

Residual categories are equivalent to twisted derived categories under certain conditions.

Provides geometric descriptions via hyperbolic reduction and Hilbert schemes.

Applications to complete intersections of quadrics.

Abstract

We show that the residual categories of quadric surface bundles are equivalent to the (twisted) derived categories of some scheme under the following hypotheses. Case 1: The quadric surface bundle has a smooth section. Case 2: The total space of the quadric surface bundle is smooth and the base is a smooth surface. We provide two proofs in Case 1 describing the scheme as the hyperbolic reduction and as a subscheme of the relative Hilbert scheme of lines, respectively. In Case 2, the twisted scheme is obtained by performing birational transformations to the relative Hilbert scheme of lines. Finally, we apply the results to certain complete intersections of quadrics.